Dispersal behaviour in cooperatively breeding mammals often differs between the sexes, which can affect how individuals of both sexes compete for breeding opportunities. However, it is largely unknown how the males and females in social mole‐rats differ in frequency, timing and social context of dispersal. Here we show, in Damaraland mole‐rats (Fukomys damarensis), that rainfall increases dispersal probabilities in both sexes. Dispersal is male biased with males dispersing earlier and more frequently in dispersal coalitions than females. Most non‐reproductive individuals disperse from the natal groups before reproducing. Territory inheritance is rare, but when it occurs, female non‐breeders inherit the breeding position from the precious breeding female. After dispersing from the natal group, males are more likely than females to join other established groups and to replace the resident breeder. Our study suggests that differences in dispersal strategy may generate contrasts in intra‐sexual competition, where male breeders are more often challenged by competitors from outside the group and female breeders may face higher competition from individuals within the group.